L(s) = 1 | + (−0.569 − 1.29i)2-s + (1.47 + 0.908i)3-s + (−1.35 + 1.47i)4-s + (0.335 − 2.42i)6-s + 2.50·7-s + (2.67 + 0.908i)8-s + (1.35 + 2.67i)9-s − 3.36·11-s + (−3.33 + 0.948i)12-s + 3.70i·13-s + (−1.42 − 3.24i)14-s + (−0.350 − 3.98i)16-s + 7.63·17-s + (2.69 − 3.27i)18-s − 0.440i·19-s + ⋯ |
L(s) = 1 | + (−0.402 − 0.915i)2-s + (0.851 + 0.524i)3-s + (−0.675 + 0.737i)4-s + (0.136 − 0.990i)6-s + 0.948·7-s + (0.947 + 0.321i)8-s + (0.450 + 0.892i)9-s − 1.01·11-s + (−0.961 + 0.273i)12-s + 1.02i·13-s + (−0.382 − 0.868i)14-s + (−0.0876 − 0.996i)16-s + 1.85·17-s + (0.635 − 0.771i)18-s − 0.100i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40500 - 0.131196i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40500 - 0.131196i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.569 + 1.29i)T \) |
| 3 | \( 1 + (-1.47 - 0.908i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 + 3.36T + 11T^{2} \) |
| 13 | \( 1 - 3.70iT - 13T^{2} \) |
| 17 | \( 1 - 7.63T + 17T^{2} \) |
| 19 | \( 1 + 0.440iT - 19T^{2} \) |
| 23 | \( 1 + 5.17iT - 23T^{2} \) |
| 29 | \( 1 - 2.27iT - 29T^{2} \) |
| 31 | \( 1 + 3.39iT - 31T^{2} \) |
| 37 | \( 1 + 7.40iT - 37T^{2} \) |
| 41 | \( 1 - 3.07iT - 41T^{2} \) |
| 43 | \( 1 + 8.40T + 43T^{2} \) |
| 47 | \( 1 + 3.63iT - 47T^{2} \) |
| 53 | \( 1 - 2.27T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 + 5.45T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 1.29iT - 73T^{2} \) |
| 79 | \( 1 + 5.01iT - 79T^{2} \) |
| 83 | \( 1 - 1.81iT - 83T^{2} \) |
| 89 | \( 1 - 5.35iT - 89T^{2} \) |
| 97 | \( 1 - 11.1iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.55039072769002572594853450542, −10.60340574714303154137557422554, −9.963705125901705248418387671099, −8.950881349466949731165198642111, −8.124894113276663632618168397358, −7.45746246412674573092678326418, −5.21316722219996264181985127274, −4.29503747168538328020321473575, −3.04058434144265175249116528068, −1.81490191998263523169668748656,
1.37441001210487316748654423987, 3.25421294065577847246175339931, 4.96812916370764200795665061575, 5.86789273526729833895989303185, 7.39351886798640496302036645598, 7.88085538123333141857394224549, 8.489142487832630410524973319238, 9.780255952694504916575362049783, 10.43356937030756357049833625802, 11.89078692231692264209691421592